Non-equilibrium coagulation processes and subcritical percolation on evolving networks

TL;DR

This paper studies non-equilibrium percolation on evolving networks, revealing critical phenomena, power-law distributions, and unique dynamic behaviors near the percolation threshold, using advanced probabilistic tools.

Contribution

It develops new analytical tools to prove critical phenomena and power-law behaviors in growing networks, specifically for the uniform attachment model, advancing understanding of non-static percolation.

Findings

Existence of a critical exponent ppa(\u03c0) for component sizes

Almost sure convergence of scaled component sizes to positive random variables

Bounded susceptibility as  approaches from below

Abstract

We investigate percolation on growing networks where the evolution of connected components resembles a non-equilibrium version of the multiplicative coalescent. The supercritical $\pi> \pi_c$ regime for a host of such models was conjectured in statistical physics, and then rigorously proven in mathematics, to exhibit behavior similar to the BKT infinite-order phase transition as $\pi\searrow \pi_c$. It has further been conjectured that the entire regime $\pi<\pi_c$ for such growing networks are ''critical'' with power-law cluster size distributions having a non-universal exponent for all values of $\pi \in (0, \pi_c)$. In this paper, we study percolation on the uniform attachment model, as a concrete template in order to develop general tools based on stochastic approximation, local convergence, branching random walks and tree-graph inequalities to prove the above conjectured phenomena. For each $\pi \in (0,\pi_c)$, we show there exists an explicit $\alpha(\pi) \in (0,\tfrac{1}{2}) $ such that the maximal component size, as well as the size of the component containing any fixed vertex, all re-scaled by $n^{\alpha(\pi)}$, converge almost surely to strictly positive random variables as the network size $n \to \infty$. These dynamics lead to novel phenomena, compared to classical 'static' models, including long-range dependence and fixation of the identity of the maximal component, within finite time, among a finite number of 'early' components. Moreover, in contrast with most static network models, we show that the susceptibility, that is, the expected size of the component of a uniformly chosen vertex, remains bounded as the network grows and $\pi$ approaches $\pi_c$ from below. The general tools developed in this paper will be used in follow-up work to understand percolation for general growing network evolution models.